A series of experiments was performed on a square silicon nitride membrane with a thickness of \({92}{}\)  and an area of [product-units=power]2 x 2 that was suspended on a silicon chip, see Figs. REF (a,b). The chip was mounted on a piezo-actuator inside a vacuum chamber at a pressure of 2 (see Methods). The motion in the center of the membrane was recorded by Doppler laser vibrometry. The measurement setup was sensitive enough to characterize the resonator's thermomechanical displacement noise in the absence of piezo-actuation, whose power spectral density (PSD) spectrum is shown in Fig. REF (c). The figure shows a peak at 188 kHz that corresponds to the fundamental vibration mode of the membrane. The peak, with an amplitude \(6.3\)  times larger than the measurement noise floor allowed to extract the linewidth. The parameters defining the linear resonance, listed in Table REF , were obtained by fitting the PSD data to a harmonic oscillator model [1]}. Then, upward frequency sweeps were performed around the found resonance with different actuation forces, with the results plotted in Fig. REF (d). As the actuation force increases, a hardening effect becomes evident from the bending of the resonance peak to the right. For an actuation force of 4.4 and above, a sudden jump in the displacement amplitude is observed, which indicates that the critical amplitude has been surpassed, i.e. the resonator shows bifurcation [2]}. The displacement maxima with amplitude \(a_p\)  for each actuation level were fitted to the backbone curve of Duffing resonators [3]},
\(\omega _p=\omega _0 \left( 1+\frac{3}{8}\gamma a_p^2 \right),\) 
